COOL CORE BIAS IN SUNYAEV–ZEL’DOVICH GALAXY
CLUSTER SURVEYS
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Citation
Lin, Henry W., Michael McDonald, Bradford Benson, and Eric
Miller. “COOL CORE BIAS IN SUNYAEV–ZEL’DOVICH
GALAXY CLUSTER SURVEYS.” The Astrophysical Journal 802,
no. 1 (March 18, 2015): 34. © 2015 The American Astronomical
Society
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http://dx.doi.org/10.1088/0004-637x/802/1/34
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Detailed Terms
The Astrophysical Journal, 802:34 (8pp), 2015 March 20
doi:10.1088/0004-637X/802/1/34
© 2015. The American Astronomical Society. All rights reserved.
COOL CORE BIAS IN SUNYAEV–ZEL’DOVICH GALAXY CLUSTER SURVEYS
Henry W. Lin1, Michael McDonald2, Bradford Benson3,4,5, and Eric Miller2
1
Harvard University, Cambridge, MA 02138, USA; henrylin@college.harvard.edu
Kavli Institute for Astrophysics and Space Research, MIT, Cambridge, MA 02139, USA
3
Fermi National Accelerator Laboratory, Batavia, IL 60510-0500, USA
4
Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
5
Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
Received 2014 July 9; accepted 2015 January 18; published 2015 March 18
2
ABSTRACT
Sunyaev–Zel’dovich (SZ) surveys find massive clusters of galaxies by measuring the inverse Compton scattering
of cosmic microwave background off of intra-cluster gas. The cluster selection function from such surveys is
expected to be nearly independent of redshift and cluster astrophysics. In this work, we estimate the effect on the
observed SZ signal of centrally peaked gas density profiles (cool cores) and radio emission from the brightest
cluster galaxy by creating mock observations of a sample of clusters that span the observed range of classical
cooling rates and radio luminosities. For each cluster, we make simulated SZ observations by the South Pole
Telescope and characterize the cluster selection function, but note that our results are broadly applicable to other
SZ surveys. We find that the inclusion of a cool core can cause a change in the measured SPT significance of a
cluster between 0.01 and 10% at z > 0.3, increasing with cuspiness of the cool core and angular size on the sky of
the cluster (i.e., decreasing redshift, increasing mass). We provide quantitative estimates of the bias in the SZ
signal as a function of a gas density cuspiness parameter, redshift, mass, and the 1.4 GHz radio luminosity of the
central active galactic nuclei. Based on this work, we estimate that, for the Phoenix cluster (one of the strongest
cool cores known), the presence of a cool core is biasing the SZ significance high by ∼6%. The ubiquity of radio
galaxies at the centers of cool core clusters will offset the cool core bias to varying degrees.
Key words: cosmology: observations – galaxies: clusters: general – galaxies: clusters: intracluster medium
estimates. Finally, contamination from radio-loud active
galactic nuclei (AGNs), generally located in the brightest
cluster galaxy (BCG), could introduce bias by “filling in” the
SZ decrement (Sayers et al. 2012).
Cool core bias (Eckert et al. 2011) is also of particular
astrophysical interest, as estimates of the cool core fraction as a
function of redshift might lead to valuable insights into the
cooling flow problem (see review by Fabian 1994). Recent
studies have found evidence for a lack of dense, cool cores in
the centers of high-redshift galaxy clusters (e.g., Vikhlinin
et al. 2007; Santos et al. 2010; McDonald 2011; McDonald
et al. 2013). These results may be evidence for an evolution in
the cooling/feedback balance, or may be a result of biases in Xray and optical surveys (e.g., Semler et al. 2012). Thus, in
order to fully understand the evolution of cooling and heating
processes in the cores of galaxy clusters, we must understand
how our sample selection affects the observed cool core
fraction.
Throughout this paper, we take the fractional matter density
Ω m = 0.27 and the fractional vacuum density ΩL = 0.73, and
neglect radiation Ωr and curvature Ω k . We introduce two
dimensionless
forms
of
the
Hubble
parameter,
h º H0 (100 km s-1 Mpc-1) = 0.7 and E (z ) º H (z ) H0 . We
define the cluster radius r500 such that the average matter
density interior to r500 is 500 times the critical density rcr of the
4
3
universe as well as the cluster mass M500 = 3 πr500
´ 500rcr .
1. INTRODUCTION
Galaxy clusters are potentially powerful tools to study dark
energy and cosmology (see review by Allen et al. 2011);
whereas standard candles test cosmology on homogeneous
scales, clusters trace the growth of inhomogeneity via N (M , z ),
the cluster number density at a given mass and redshift.
Accurate cluster cosmology requires not only reliable estimators of clusters properties such as total mass (i.e., M500), but
also an accurate estimation of the survey selection function.
A new technique of assembling a nearly unbiased sample of
galaxy clusters uses the Sunyaev–Zel’dovich (SZ) effect
(Sunyaev & Zeldovich 1972): the inverse Compton scattering
of photons as they pass through the hot intracluster medium,
leading to distorted-spectrum patches in the cosmic microwave
background (CMB). One attractive characteristic of SZ surveys
is that the SZ signal is virtually redshift independent, as
opposed to the ~1 d L2 flux dimming in infrared, optical, and Xray surveys. Recently, large samples (100) of galaxy clusters
have been assembled using SZ selection that include clusters
out to z ~ 1.5 (e.g., Vanderlinde et al. 2010; Ade et al. 2011;
Hasselfield et al. 2013; Reichardt et al. 2013) including many
of the most massive known galaxy clusters (e.g., Foley et al.
2011; McDonald et al. 2012; Menanteau et al. 2012).
Many potential systematics have been considered in SZselected cluster surveys. Simulations have shown that SZ
surveys are expected to be relatively insensitive to the effects of
non-gravitational physics (Nagai 2006), projection effects
(Shaw et al. 2008), and the dynamical state of the cluster
(Krause et al. 2012; Battaglia et al. 2012). The presence of a
“cool core”—a central region of over-dense gas accompanied
by a drop in temperature—could also conceivably bias mass
2. METHODS
In order to measure the influence of cool cores and radioloud AGNs on the SZ signal, we generate one-dimensional
(spherically symmetric) mock galaxy clusters with properties
1
The Astrophysical Journal, 802:34 (8pp), 2015 March 20
Lin et al.
Table 1
Parameters
Parameter
Range
Distribution
Notes
M500 M
z
β
ϵ
rc r500
log rs r500
[1 ´ 1014, 2.1 ´ 1015]
[0, 2]
(0.28, 0.88, 0.71)
7.1 - 6b b
(0.067, 0.26, 0.47)
1.21
Log-uniform
Uniform
Trianglea
L
Trianglea
L
SPT 2500 deg2; Bleem et al. (2014)
SPT 2500 deg2; Bleem et al. (2014)
McDonald et al. (2013), Vikhlinin et al. (2006)
Andersson et al. (2011)
McDonald et al. (2013)
McDonald et al. (2013)
Note. All radii are in units of Kiloparsecs. We report 1s uncertainties.
a
We fit triangle distributions to the empirical distributions derived from McDonald et al. (2013) to avoid the long Gaussian tails that lead to unphysical clusters. We
report the (min, max, mode) of the distribution in the range column.
b
The mock SZ signal is virtually ϵ-independent. We choose ϵ to let d log P d log r = -0.90 at large radii, as given by the universal temperature profile of Arnaud
et al. (2010).
For a pure non-cool core, a = 0 and Equation (1) reduces to
the simpler form
drawn from existing samples of high-mass, relaxed clusters,
and add a perturbation either to the gas density (cool core) or
mm flux (radio source). Simulated observations of these mock
clusters with the South Pole Telescope are generated, including
realistic noise and background, yielding a realistic SZ signal-tonoise measurement.
n p ne =
Since we are mostly interested in the SZ signal of our mock
clusters, the key ingredient in our cluster models is the pressure
profile P(r). However, we must also model the cluster density
profiles r (r ) since we wish to calculate the change in the
pressure profile dP due to a density perturbation dr , which in
general is not dP µ dr as the temperature profile also shifts
T  T - dT . Consequently, we need an additional constraint
on our clusters to calculate dP , which we take to be hydrostatic
dP
dF
equilibrium: dr = -r dr , where Φ is gravitational potential.
Since there is no evidence that the underlying dark matter
distribution is affected by the baryonic processes driving cool
cores (Blanchard et al. 2013), dF is straightforward to
calculate, and using P (r500 ) as a boundary value, it is possible
to calculate P + dP .
The details of our density and pressure parameterizations are
now presented. For each non-cool core cluster, we draw a
cluster mass M500 and a redshift from uniform distributions
(see Table 1), with ranges motivated by the observed ranges in
the SPT 2500 deg2 survey (Bleem et al. 2014). The gas density
profiles were then modeled by a slightly modified version of
the functional form of Vikhlinin et al. (2006):
-a
n 02 ( r rc + d )
2 ù 3b - a 2
é 1 + ( r rc )
ëê
ûú
1
é 1 + ( r rs )3ù
úû
ëê
3
,
1
2 ù 3b
é 1 + ( r rc )
ëê
ûú
é 1 + ( r rs )3ù
úû
ëê
3
,
(2)
where α, β, and ϵ are slope parameters for the r  rc ,
rc  r  rs and r  rs radial regimes. We convert n p ne to the
gas mass density via the relation r = m p ne A Z , where
Z = 1.199 is the average nuclear charge, which is greater than
unity because of ionized elements heavier than hydrogen,
A = 1.397 is the average nuclear mass, which is greater than Z
due to the presence of neutrons, and ne = Zn p is the electron
density in terms of the positive nuclei density np. The density
normalization n0 was chosen to enforce an average gas fraction
fgas = 0.125 within r500.
For the unperturbed, non-cool core clusters, we adopt the
simple temperature profile
2.1. Constructing Mock Galaxy Clusters
n p ne =
n 02
-0.45
é
æ r ö2 ùú
T (r )
ê
÷
,
= 1.35 ê 1 + çç
÷
ççè 0.6 r500 ÷÷ø úú
T0
êë
û
(3)
which is the “universal” temperature profile of Vikhlinin et al.
(2006) with the term accounting for the central temperature
drop suppressed.
For each non-cool core cluster, we generate 14 progressively
cuspier cool-core clusters. We start by duplicating each noncool core cluster and shifting r  r + dr . To avoid adding
more free parameters (a cooling radius, etc) which may be
correlated with the other model parameters, we simply
increment α to steepen the density profile toward the center
of the cluster. Each successive cluster has an = an - 1 + 0.25.
The density normalization n0 was also reduced by an
appropriate factor to keep the gas mass within r200 constant.
Finally, the assumption of hydrostatic equilibrium was used
to recalculate the pressure profile of the cool core cluster
P  P + dP :
(1)
where the various length scales and slopes were sampled from
realistic ranges for massive galaxy clusters by a Monte Carlo
process, which we detail in Table 1. For this work, we have
“regulated” the parameterization of Vikhlinin et al. (2006) by
inserting a small factor d = 0.1 to obtain a finite density at
r = 0. This is necessary as we will eventually extract an SZ
signal from the entire cluster, whereas the behavior of the
profile very near r = 0 is not of interest to observational works
due to the limited spatial resolution of any survey.
P + dP =
é r + dr dP
(r + dr ) G dM ùú
.
úû
r
dr˜
r˜2
ò dr˜ êêë
(4)
We vary P (r200 ) until the “internal energy” µ ò PdV of the cool
core cluster out to r200 agrees with its non-cool core
counterpart. Note that in general, this means that
dP (r200 ) ¹ 0, though at r200, dP  P .
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The Astrophysical Journal, 802:34 (8pp), 2015 March 20
Lin et al.
Figure 1. Clockwise from the upper left: density, temperature (Tx = ò n 2 T dV ò n 2 dV ), cooling time, and entropy as a function of cluster radius for a typical mock
cluster. All of the displayed profiles were obtained by cloning a single non-cool core core cluster and adding density perturbations. These plots represent 1/200th of the
total dataset. In the upper right plot, the thick dashed line represent the universal temperature profile of Vikhlinin et al. (2006). In the lower left plot, the thick dashed
line represents the predicted entropy profile slope from non-radiative simulations of Voit et al. (2005).
(2013), the cool core clusters have a central entropy
K0  100 keV cm2 , whereas for non-cool core clusters
K0  100 keV cm2 . The slope of our entropy profiles
outside of the core is also in reasonable agreement with
the accretion shock model of Voit et al. (2005), which
represents the “baseline” entropy profile of clusters if
radiative and other non-gravitational processes are
ignored.
3. Cooling Time. The cooling time tcool has been shown by
Hudson et al. (2010) to be a quantity that can be used to
segregate cool cores from non-cool cores. In Figure 1(d),
tcool = 3kT ( ne L) is plotted as a function of radius. Here,
L (T , Z ) is the cooling function given by Sutherland &
Dopita (1993). Most non-cool cores have central cooling
times tcool,0  1 H0 whereas the cool core clusters have
cooling times tcool,0 ~ 1 Gyr , consistent with the findings
of Hudson et al. (2010).
The above methods were used to generate 200 × 15 mock
clusters which ranged from pure non-cool core a = 0 to strong
cool core a = 3.5. We chose the maximum value a = 3.5 as
even strong cool core clusters like Phoenix have a < 3.5.
2.2. Model Validation
Additional quantities were derived to demonstrate how the
wide range of observed cluster properties are captured by our
methodology.
1. Temperature. Our derived temperature profiles are
displayed in Figure 1(b). The cool core profiles exhibit
a central temperature drop similar to the average cool
core profiles observed in X-ray selected samples
(Vikhlinin et al. 2006). The characteristic temperatures
of our clusters range from ~1 to ~30 keV, which is in
agreement with observations of massive clusters. Furthermore, Tmin T0 , which corresponds to the ratio of the
central cool core core temperature to the a = 0 central
temperature, range from 0.1 to 0.85 in the (Vikhlinin
et al. 2006) clusters, which is completely covered by our
simulated clusters. Similarly, our simulations cover the
range of temperature profiles observed in (McDonald
et al. 2014), which find that the fractional temperature in
0.09
the inner 0.04R500 is Tcore T500 = 0.740.04 for high
redshift, cool core clusters and higher for low redshift
cool core clusters.
2. Entropy Parameter. The entropy profiles of our mock
clusters are derived in an effort to show their resemblance
to observed clusters. Following the X-ray survey
conventions, an entropy parameter K º k B Tne-2 3 is
introduced. We plot this parameter as a function of
radius in Figure 1(c). Consistent with Cavagnolo et al.
(2009), Hudson et al. (2010) and McDonald et al.
2.3. SZ Maps
For each mock cluster, we constructed an SZ map with an
angular resolution of Dq = 0.125 arcmin , which was converted to spatial resolution Dr2D = d A (z ) Dq where dA(z) is the
angular diameter distance to the cluster. The SZ line of sight
integral was then calculated at each pixel:
DT TCMB =
ì
ï
òq dl ïíïï fsz (n , T )
î
ï
st n e k B T ü
ï
ý
2 ï
me c ï
þ
æ eX + 1
ö
fsz (n , T ) = çç X
- 4÷÷÷ éë 1 + dsz ( X , T ) ùû
ççè e X - 1
÷ø
(5)
(6)
where X º hn k B TCMB, st is the Thomson cross section, me is
the mass of the electron, c is the speed of light, and dsz is the
3
The Astrophysical Journal, 802:34 (8pp), 2015 March 20
Lin et al.
Figure 2. Dependence of the SZ bias on the cuspiness (g º - d log r ), redshift (z), and angular size (Q500 º r500 d A (z )) for 3000 mock clusters. In the left panel, the
d log r
point color corresponds to redshift of the mock cluster. In the central panel, we color by the angular size of the cluster. The clear separation into colored bands in this
plot suggests that the bias is more fundamentally tied to the angular size than it is to the redshift. In the right-most panel, we show the edge-on projection of the best-fit
three-dimensional plane (Equation (7)). The ∼0.5 dex scatter is primarily due to variations in the cluster gas density profile, and the imperfect classification of cool
core strength based on the γ parameter.
relativistic correction as given by Itoh et al. (1998). For our
purposes, it sufficed to evaluate dsz (T ) at T (r500 ). We made
simulated SZ maps at 97.6 and 152.9 GHz, corresponding to
the effective frequency of the SPT observing bands used for
cluster-finding (Carlstrom et al. 2011). Our mock SZ maps
were generated at a resolution 1¢ and then later convolved
with the SPT beam which has an effective FWHM of ~1.2
¢ .
The mock SZ maps were then spatially filtered in a way
identical to the SPT cluster-finding algorithm used in Reichardt
et al. (2013). A multi-frequency matched spatial filter
(Haehnelt & Tegmark 1996; Melin et al. 2006) was applied
to the SZ maps in Fourier space, which accounted for the
cluster gas profile, and the other sources of astrophysical signal
and instrumental noise expected in the SPT maps. The spatially
filtered signal at the true cluster position was then renormalized so that it was equivalent to the unbiased SPT
significance, ζ, used in Vanderlinde et al. (2010), which would
effectively correspond to the signal-to-noise of cluster in a
SPT map.
These SZ maps were run through the SPT cluster-finding
pipeline, which adds noise in the form of point sources and
CMB anisotropy maps, before attempting to detect clusters. For
each cluster, the SPT pipeline then reported ζ, the “unbiased
significance” (see Vanderlinde et al. 2010; Benson et al. 2011)
that roughly corresponds to the cluster signal-to-noise ratio.
We convert these luminosities to the SPT observing
frequencies by assuming a spectral slope of as = 0.89, where
flux scales with n as . This slope was the median spectrum
between 1.4 and 30 GHz for radio galaxies in a sample of 45
massive clusters observed by Sayers et al. (2012), however it is
always possible to reinterpret the results for different choices
slopes. We discuss deviations from this assumption in
Section 3.1. As a control, a point source is not added to one
cool-core/non-cool core cluster at each mass and redshift.
3. RESULTS
In Figure 2, we show the fractional change in the SPT
detection significance as a function of the cool core cuspiness
(Vikhlinin et al. 2007), angular size (Q500 º r500 d A (z )), and
redshift. At Q500 < 3¢, the SPT detection significance changes
by Dz z = (z CC - z NCC ) z NCC  0.1 for any type of cool core
at any redshift. The cool core bias is most pronounced for lowredshift, high mass systems, where the angular size of the SPT
beam is much smaller than the angular size of the cluster so that
the inner pressure profile becomes important. In principle, the
resulting redshift-dependence in the bias might lead to a ~2%
bias toward cool core clusters from z ~ 1.5 to z ~ 0.3 in the
observed cool core fraction CCF º Ncool core Nclusters evolution,
but these effects should be negligible when compared to the
~30% evolution in CCF reported in e.g., McDonald
et al. (2013).
In real observational studies it may be difficult to determine
the density shape parameter α, as this requires a multiparameter fit to the n p ne profile and assumptions on the shape
of the cool core/non-cool core profiles. To ameliorate the
situation, we calculate the “cuspiness” parameter
g º -d log r d log r ∣r = 0.04r500 (Vikhlinin et al. 2007) of all
simulated clusters. Since this quantity does not refer to the
details of our parameterizations, it serves as a modelindependent indicator of cool core strength; correlations
between these quantities and Dz z are shown in Figure 2.
The bias is well-captured by the following relation:
2.4. Active Galactic Nuclei
In order to test the effects of including radio-loud AGNs, we
created mock clusters with the following grid of parameters:
1. M500 2 ´ 1014 M = 10 0 , 101 2 , 101;
2. z = 0.3, 0.4, 0.5, 0.7, 1.1, 1.4, 1.7;
and we include a radio-loud AGN at the cluster center,
with a range of luminosities:
3. log L1.4 = 23–29, D log L1.4 = 1.0 ,
where L1.4 is the 1.4 GHz luminosity in WHz-1. All other
parameters are set to their median values. The range in
log L1.4 covers the most luminous BCG in the 152 X-ray
BCG sample of Sun (2009), allowing for unusually
shallow spectrum systems that would generate a
large bias.
dz z = 10
-2.08  0.01
(Dg )
æ Q500 ( z ) ö1.52  0.02
÷÷
çç
÷
èç arcmin ÷ø÷
0.85  0.01 ç
(7)
where Dg = g - g ∣a= 0 » g - 0.1. This expression gives a
4
The Astrophysical Journal, 802:34 (8pp), 2015 March 20
Lin et al.
Figure 3. Dependency of Dz z on M500, redshift, radio luminosity, and spectral slope. The contours represent lines of constant log (Dz z ). If an AGN exists in the
shaded region, it will overpower the SZ signal. The large black dot represents a Phoenix-like cluster at multiple redshifts. Even at z = 0.3, a Phoenix-like cluster will
exhibit only a 1% bias. The most extreme systems exhibit a ~10% bias at z ~ 0.3. Red dots represent a sample of X-ray selected massive elliptical galaxies from
Dunn et al. (2010) and blue dots represent extreme AGNs hosted in massive clusters Hlavacek-Larrondo et al. (2012).
28.5 GHz. Lin et al. (2009) explicitly measure the radio
emission at frequencies ranging from 1.4 to 43 GHz in an X-ray
selected sample of clusters and find mostly steep spectra
(as > 0.5), but also find a substantial number of flat or even
inverted spectra. Neither of these studies—or any published
studies of radio galaxies in clusters, for that matter—extend to
∼100 GHz, so there is still substantial uncertainty in the value
of as that we should be using.
In Figure 4 we show the strong cool core (a = 2.75) bias
and the radio-loud AGN bias as a function of redshift.
Interestingly, while SZ surveys are biased toward cool cores,
they are biased against radio-loud AGNs, and this bias is of the
same order of magnitude as the strong cool core bias, defined
as the average Dz z over cool cores with a = 2.75. Since
powerful radio galaxies tend to live in the center of stronglycooling galaxy clusters, the net bias is partially canceled (see
Figure 4). In principle, low-redshift galaxy clusters that harbor
a powerful, radio galaxy (log L1.4  25) but lack a cool core
will be under-represented in SZ surveys. However in practice,
such systems appear to be uncommon in nature (see e.g.,
Sun 2009; Dunn et al. 2010). The AGN bias is more strongly
redshift dependent than the cool core bias due to the ~1 d L2
flux dimming where dL is the luminosity distance, whereas the
evolution in the simulated cool core bias is a result of SPT
beam effects and weak dependencies of our parameterizations
on rcr (z ).
practical method for estimating the SZ bias of a given cool core
cluster, assuming that z > 0.1.
3.1. Radio Bias
Assuming a typical spectral slope of as = 0.89 (Sayers
et al. 2012), only sources with log L1.4 > 26 result in
∣ Dz z ∣  10% for z > 0.3 and M > 3 ´ 1014 . Consequently,
none of the 152 radio-loud AGNs listed in Sun (2009) would
produce a bias in excess of 10% if located in massive clusters.
More generally, the radio-loud point source bias can be readily
estimated by the following fit to our mock clusters (see
Section 2.4):
-1
æ nSZ ö-as æç S1.4 ö÷ æç M500 ö÷
÷
÷ .
dz z = -0.03 çç
֍
çè 1.4 GHz ÷÷ø ççè mJy ÷÷ø ççè 1014 M ÷÷÷ø
(8)
In Figure 3 we show a contour plot of Dz z on
Q500 (M500 , z ), a, L1.4 parameter space. We display flux and
luminosity data from a sample of X-ray selected massive
elliptical galaxies from Dunn et al. (2010) and extreme AGNs
hosted in massive clusters Hlavacek-Larrondo et al. (2012). As
can be seen by Figure 3, the fractional change in SPT
significance will be strongly dependent on the assumed value
of as , which has a fairly large scatter. For example, Coble et al.
(2007) measures a median value of as = 0.72 between 1.4 and
5
The Astrophysical Journal, 802:34 (8pp), 2015 March 20
Lin et al.
thermal pressure, non-thermal pressure will persist for on
roughly the dynamical timescale (Shi & Komatsu 2014).
From Equation (8) we see that bias scales as 1 M , so to first
order, we expect that adding non-thermal pressure will rescale
the bias by (1 + Pnon - therm Ptherm )-1, if we assume that the
processes responsible for forming cool cores and providing
non-thermal pressure are uncorrelated. As clusters evolve, this
term could introduce an additional redshift dependence on the
cool core bias, if the processes generating non-thermal pressure
evolve with redshift.
4.2. Varying fgas
Throughout this work, we have assumed that fgas = 0.125
with no scatter. However, the numerical simulations of
Battaglia et al. (2014b) which included radiative physics and
AGN feedback estimate a scatter ~10% in fgas at r < r500 .
Eckert et al. (2013) suggest that cool core clusters and non-cool
core clusters differ in fgas by a few percent, with cool core
clusters more reliably tracing the cosmic baryon fraction.
For a fixed number of baryons, a lower fgas will result in an
increase in the pressure normalization, in order to overcome the
steeper potential due to additional dark matter. For a cluster in
giving
hydrostatic
equilibrium,
kT µ M r
23
, which means that the effects
P µ M 2 3 ~ (1 fgas )2 3Mbaryon
of changing 1 fgas will be roughly equivalent to changing M500 .
Equations (7) and (8) can therefore be rescaled by a factor
(fgas 0.125). Adding intrinsic scatter in fgas would then simply
add scatter in the SZ bias. If fgas varied with redshift, this could
lead to a further evolutionary factor in Equation (8). However,
Battaglia et al. (2013) found little evidence for such a redshift
dependence, as has long been assumed (White et al. 1993).
Figure 4. Strong cool core (a = 2.75) bias and radio-loud AGN bias. The
black, dotted lines represent the massive cool core “Phoenix” cluster, with
radio spectral slope as = 1.3 (McDonald et al. 2014) and X-ray properties
given in McDonald et al. (2013), for a range of redshifts. The pink shaded
region represents a series of radio-loud AGNs with spectral slope as ⩾ 0 ,
which demonstrates the range in possible AGN bias. The dark blue lines are the
median bias in each bin and the light blue regions represent 2s confidence
intervals. In general, the cool core bias and the radio-loud AGN bias work to
roughly cancel each other.
4. ROBUSTNESS OF SIMULATIONS
In this section we estimate how relaxing several of our
simplifying assumptions could change our bias estimates.
Though a thorough treatment of each of these issues is beyond
the scope of this paper (and should be addressed by, e.g., Nbody simulations), an estimate on the relative importance of
these effects can be obtained by recasting these effects into an
M500 dependence.
4.1. Deviations from Hydrostatic Equilibrium and Non-thermal
Pressure
4.3. Effects and Evolution of AGNs
We purposefully exclude any evolution in AGN properties
as a function of cluster mass and redshift, despite evidence that
such links exist (e.g., Ma et al. 2013). However, by casting
Equation (8) in terms of the cluster mass, redshift, and radio
luminosity, we allow the direct incorporation of such trends
into the bias estimate. By fully sampling a grid of parameters
(see Section 2.4), we make certain that, regardless of how
AGNs evolve, we understand how they influence the SZ signal
at all redshifts and cluster masses.
Radio-mode AGN feedback can modify the gas distribution
of galaxy clusters as gas is heated and expelled from the core,
leading to deviations from hydrostatic equilibrium in the inner
region. Such processes will typically result in less pressure than
predicted by hydrostatic equilibrium, since the gas is supported
by feedback in addition to thermal pressure.
In the absence of complicated astrophysics, a galaxy cluster
will
equilibrate
on
dynamical
timescales
tdynamic ~ R500 v ~ (Gr )-1 2 ~ 10 9 yr. For hydrostatic equilibrium to generically hold, the cooling timescale should be
much longer than the dynamical timescale tcool  tdynamic so
that hydrostatic equilibrium will be restored efficiently when
gas cools. If the cooling timescale is shorter than the
equilibrium timescale in the core of the cluster, we would
expect pressure to be lower than what is required to maintain
hydrostatic equilibrium, since gas must be sinking toward the
core. This would mean that the bias should be generically lower
than what we have derived, since the observed change in
pressure should be smaller than that required to maintain
hydrostatic equilibrium dPobs ⩽ dPeq .
Throughout our discussion, we have assume that pressure is
purely thermal, e.g., P = nkT . Analytic work (Shi &
Komatsu 2014), numerical simulations (Dolag et al. 2005;
Sembolini et al. 2013; Battaglia et al. 2012; Nelson et al. 2014),
and multi-wavelength observations (Ade & Aghanim 2013;
Donahue et al. 2014; von der Linden et al. 2014; Sereno
et al. 2014), however, have shown that non-thermal pressure
can contribute significantly (~20%; increasing with cluster
radius), which can lead to an underestimate of M500 by about
~10% (Shi & Komatsu 2014). Sources of non-thermal
pressure include turbulence from intracluster shocks, magnetic
fields, and cosmic rays. If turbulence dominates the non-
5. APPLICATION TO WELL-KNOWN SYSTEMS
Using Equations (7) and (8), we can now calculate how
biased SZ surveys are (or are not) for some well-studied
extreme systems. First, we consider two of the strongest known
cool cores: Abell 1835 (g = 0.85, M 500 ~ 1015 M; McNamara et al. 2006) and the Phoenix cluster (g = 1.29,
M500 ~ 1.3 ´ 1015 M; McDonald et al. 2012). Assuming
both of these clusters are at z = 0.6, we find Dz z = 3.6% and
5.9%, for Abell 1835 and Phoenix, respectively. At z = 1.0,
these biases are reduced to 2.8 and 4.2% (see also Figure 4).
6
The Astrophysical Journal, 802:34 (8pp), 2015 March 20
Lin et al.
Given that these are two of the most extreme cool cores known,
we expect the typical bias toward selecting cool cores in SZ
surveys to be 5% at z > 0.5.
Figure 3 quantifies the radio bias for a variety of nearby
clusters from Dunn et al. (2010) and Hlavacek-Larrondo et al.
(2012), but here we specifically look at two well-known nearby
radio-loud central galaxies: Hydra A (S1.4 = 40.8 Jy,
z = 0.055; Bîrzan et al. 2004) and Cygnus A (S1.4 = 1600
Jy; z = 0.056; Bîrzan et al. 2004), the latter being one of the
most powerful examples of radio-mode AGN feedback known.
At z = 0.6, these two clusters would be biased low in SZ
significance by 1.8 and 110%, respectively. At z = 1.0, these
biases are further reduced to 0.5 and 31%. Thus, while SZ
surveys may miss the most extreme radio-loud clusters, this
bias is rapidly reduced with increasing redshift. In nearby Xray-selected clusters, 1% of clusters have radio luminosities
as high as Cygnus A (M. Hogan et al. 2015, in preparation), so
we expect this bias to be small overall.
the bias from radio-loud point sources should dominate any
cool core bias.
Our results support the long-asserted claim that an SZselected sample of galaxy clusters is a robust cosmological
probe: though we observe a small bias in the mass estimator,
the magnitude of the bias is much smaller than the typical
scatter in the mass relationship.
We provide estimates of the SZ bias as a function of redshift,
mass, and cuspiness, parameters that are model-independent, as
well as the radio bias. One can estimate the bias of a given
system easily by plugging in values to our function
Dz z = f (z, r500, g ) + g (L1.4, z, M500 ) where f (z, r500, g ) is
given by Equation (7) and g (L1.4, z, M500 ) is given by
Equation (8). By quantifying the cool core bias, astrophysical
constraints on cool core properties from SZ surveys can now be
more reliably interpreted. In particular, constraints on the cool
core fraction from an SZ-selected dataset are only subject to a
systematic bias of order one percent, a significant reduction
over X-ray selection. Since there is a stringent upper limit on
the redshift evolution of the cool core fraction bias, we can now
confidently say that almost all observed evolution in the cool
core fraction reflects genuine cool core evolution. With the
arrival of results from Planck, SPT, and the Atacama
Cosmology Telescope and others, SZ surveys might be ideal
for studying the mysterious balance between heating and
cooling.
6. MASS BIAS IN SZ SURVEYS
To translate SZ observations to cosmological constraints, it
is necessary to estimate mass distribution of an ensemble of
clusters. To this end, Vanderlinde et al. (2010) and Benson
et al. (2011) have adopted ζ as a proxy for M500 by assuming a
scaling relation z µ M B . As a consequence, the true M500 of
cool core clusters will be underestimated by an amount on the
order of ò P (Q500 , g ) ádz z ñ d Q500 dg on average (for B ~ 1)
if the calibration was performed using only a sample of noncool core clusters. In principle, this redshift-dependent bias
should translate into a distorted N (M , z ); however, the lognormal intrinsic scatter in ζ given mass has been measured to
be 0.21 ± 0.10, calibrated using X-ray observations (Benson
et al. 2013). Given that the Phoenix cluster has the most-cuspy
X-ray surface brightness profile of the SPT clusters with
Chandra X-ray follow-up (McDonald et al. 2013), which
includes ∼90 clusters, we expect the bias in ζ to represent an
extreme example. Therefore, the scatter in ζ given mass due to
cool cores should be a factor of several below the overall
scatter in ζ and its current measurement uncertainty. The
uncertainty in scatter has a negligible effect on the cosmological constraints from current SZ cluster surveys (e.g., Benson
et al. 2013), so the additional scatter from the effect of cool
cores will be even less significant.
Although we have focused on the SPT SZ survey, our results
should be applicable to other SZ surveys, e.g., Planck, ACT,
etc. Broadly speaking, we expect less bias in a survey with
lower angular resolution and greater bias in a higher-resolution
survey, since convolving a wide survey beam function with a
cuspy SZ signal will smooth it to a less cuspy signal.
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